# American Institute of Mathematical Sciences

July  2017, 22(5): 1965-1975. doi: 10.3934/dcdsb.2017115

## Topological stability in set-valued dynamics

 1 Instituto de Matemàtica y Ciencias Afines (IMCA), Universidad Nacional de Ingeniera Calle Los Biòlogos 245, 15012 Lima, Perù 2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530,21945-970 Rio de Janeiro, Brazil 3 Institut de Mathématiques Université de Bordeaux Ⅰ, 33405, Talence, France

* Corresponding author: Carlos Arnoldo Morales Rojas

Received  November 2016 Revised  January 2017 Published  March 2017

Fund Project: Work partially supported by CNPq from Brazil and MATHAMSUD 15 MATH05-ERGOPTIM, Ergodic Optimization of Lyapunov Exponents.

We propose a definition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the set-valued sense is topologically stable in the classical sense [14]. Next, we prove that every upper semicontinuous closed-valued map which is positively expansive [15] and satisfies the positive pseudo-orbit tracing property [9] is topologically stable. Finally, we prove that every topologically stable set-valued map of a compact metric space has the positive pseudo-orbit tracing property and the periodic points are dense in the nonwandering set. These results extend the classical single-valued ones in [1] and [14].

Citation: Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115
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